Question: Prove that if L[u] = f is a real inhomogeneous linear system with real right-hand side f, and u = v + i w is

Prove that if L[u] = f is a real inhomogeneous linear system with real right-hand side f, and u = v + i w is a complex solution, then its real part v is a solution to the system, L[v] = f, while its imaginary part w solves the homogeneous system L[w] = 0.

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